Average induced EMF and average induced current of a flipped loop?

[Violagirl]

Homework Statement

A wire loop of resistance R and area A has its normal along the direction of a uniform magnetic field, B. The loop is then flipped over in a time Δt so that its normal is opposite to the field. a) Calculate the average induced EMF. b) Find the average induced current. c) If the field is out of the page, what is the direction of the induced current? d) What is the total charge transported through the circuit in the interval Δt?

Homework Equations

I_B = ∫NBA sin θ

ε_ind = dI_B/ dt

θ = ωt

The Attempt at a Solution

I've drawn the situation out on paper. In the beginning, it seems that B and A are uniform to one another at angle of cos θ = 0 degrees, which equals 1.

After the loop is flipped, B and A are no longer uniform as A will be downwards with B in the same, original direction. They will be an angle of cos θ = 180 degrees, which equals -1.

I also know that since they are looking for the average induced EMF, I will need to use the equation:

E_ind = ΔI_B/Δt

So I will need to find the separate EMF values of each position of the loop.

I know then that for the average induced current, it will be similar in that I will need:

I_ind = ΔE_ind/ΔR

From here though, I'm confused on if I'm starting it correctly. In the first situation, I thought that B and A were uniform to one another so I took:

I_b = ∫NBA sin ωt

I_b = -NBAω cos ωt

so then I_b = NBAω

But then taking the derivative to find E_ind is where I'm having trouble. If I wait to find the derivative of -NBAω cos ωt, we get NBAω sin ωt, which give an answer of 0 since sin 0 degrees is equal to 0 and it doesn't seem right...

 

[haruspex]

E_ind = ΔI_B/Δt

So I will need to find the separate EMF values of each position of the loop.

There are not "separate EMF values of each position". There is a flux at each position. The EMF arises from the rate of change of flux.
Since you only want the average EMF, You don't need to worry about individual positions at all. ΔI_B in your equation represents the total change of flux between the two positions. (To avoid confusion with current, I'll use ##\Phi## instead.) Since you only care about the average EMF over time Δt, you can just take the difference in flux between the two positions, ##\Delta\Phi##, and divide by Δt.
What is the flux through the loop in the initial position?